Ultimately, you probably want to consider which sets are
"more like" other sets than others according to some
distance measure, then organize them
so that the sets most like each other are closest together.
You probably also want to know which elements make sets
intersect, and place them next to each other.
One way to deal with this is consider the sets as columns and the
set members as rows. A jaccard index is one distance measure
I have used for this purpose, but there are plenty of others. A
hierarchical cluster in both dimensions will place similar sets and
similar elements together.
One can visualize data organized this way using something like a
heatmap. Generally speaking, this kind of representation will
reveal the trends you are trying to get at with a high dimensional
Venn diagram.
Just a couple of thoughts.
T